Michael Reeks, Alistair Savage
To each symmetric graded Frobenius superalgebra we associate a W-algebra. We then define a linear isomorphism between the trace of the Frobenius Heisenberg category and a central reduction of this W-algebra. We conjecture that this is an isomorphism of graded superalgebras.
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We associate a graded monoidal supercategory $\mathcal{H}\mathit{eis}_{F,k}$ to every graded Frobenius superalgebra $F$ and integer $k$. These categories, which categorify a broad range of lattice Heisenberg algebras, recover many previously defined Heisenberg categories as special cases. In this way, the categories $\mathcal{H}\mathit{eis}_{F,k}$ serve as a unifying and generalizing framework for Heisenberg categorification. Even in the case of previously defined Heisenberg categories, we obtain new, more efficient, presentations of these categories, based on an approach of Brundan. When $k=0…
This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that any Frobenius category is equivalent to an extension-closed exact subcategory of the Frobenius category formed by Cohen-Macaulay modules over some additive category; this is an analogue of Gabriel-Quillen's embedding theorem for Frobenius categories; (3) we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure, with …
We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle. By putting restrictions on these points we obtain various Frobenius subcategories. The main purpose of constructing these Frobenius categories is to give a precise and elementary description of the triangulated structure of their stable categories. We show in arXiv:1209.1879 for which parameters t…
We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category $\F$ such that the homotopy category of this model structure is equivalent to the stable category $\underline{\F}$ as triangulated categories.
Foundations of Frobenius Heisenberg categories
本文研究 Frobenius Heisenberg 范畴的态射空间基，利用范畴余乘法与广义分圆商证明相关猜想，并表明其 Karoubi 包络的 Grothendieck 环恢复对应格 Heisenberg 代数。
Towards higher Frobenius functors for symmetric tensor categories
本文发展了正特征下张量范畴的单向函子理论与示例，扩展了弗罗贝尼乌斯函子，通过描述新的通用韦尔林德范畴Ver_{p^n}的构造，探讨其在分类中等增长张量范畴中的潜力。
Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in p…
phantom stable categories of $n$-Frobenius categories are triangulated
本文研究n - 弗罗贝尼乌斯范畴的幻稳定范畴，通过由n - 投射对象构造合冲函子，证明其不仅是加法函子还能诱导自等价函子，最终得出幻稳定范畴是三角化的，平移函子由该自等价函子担任。
We introduce formulae of Frobenius-Schur indicators of simple objects of Tambara-Yamagami categories. By using techniques of the Fourier transform on finite abelian groups, we study some arithmetic properties of indicators.
Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories as higher dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of $n$-exangulated categories contains $n$-exact categories and $(n+2)$-angulated categories as examples. In this article, we introduce a notion of Frobenius $n$-exangulated categories which are a generalization of Frobenius $n$-exact categories. We show that the stable category of a Frobenius $n$-exangulated category is an $(n+2)$-angulated category. As an application, this result generalizes the work by Jasso. We provide a c…
We give two alternate presentations of the Frobenius Heisenberg category, $\mathcal{Heis}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus\dotsb\oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of $\mathcal{Heis}_{F,k}$ are given in terms of planar diagrams consisting of strands "colored" by integers $i=1,\dotsc,n$, where a strand of color $i$ carries tokens labelled by elements of $F_i.$ In addition, we prove that when $F$ decomposes this way, the tensor product of Frobenius Heisenberg categories, $\mathcal…
Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we first introduce the concept of left Frobenius pairs on an extriangulated category C, and then establish a bijective correspondence between Frobenius pairs and certain cotorsion pairs in C. As an application, some new admissible model structures are established from left Frobenius pairs under certain conditions, which generalizes a result of Hu et al. (J. Algebra 551 (2020) 23-60).
Frobenius quotients, inflation categories and weighted projective lines
本文提出Frobenius商的概念，并展示其在膨胀范畴中的应用，通过向量丛在加权射影线上诱导特定的Frobenius商。
We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k…
Leclerc recently studied certain Frobenius categories in connection with cluster algebra structures on coordinate rings of intersections of opposite Schubert cells. We show that these categories admit a description as Gorenstein projective modules over an Iwanaga-Gorenstein ring of virtual dimension at most two. This is based on a Morita type result for Frobenius categories.
Given a tensor functor between tensor categories $\mathcal{C}$ and $\mathcal{D}$, we give criteria that, under certain assumptions, the Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of the other one. We also give an affirmative answer to Natale's question asking if the class of Frobenius tensor categories is closed under exact sequences.
Affine Frobenius Brauer Categories
本文定义了与对称对合 Frobenius 超代数相关联的仿射 Frobenius Brauer 范畴，构造了其对正交辛李超代数超模范畴的作用，并提出同态空间基的猜想及证明思路。
We construct continuous Frobenius categories of type $D$. The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. Discrete laminations are clusters. This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$.
We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.
Phantom stable category of $n$-Frobenius categories
本文定义了 $n$-Frobenius 范畴的幻影稳定范畴，证明其存在性，并研究其性质，推广了 Frobenius 范畴的稳定化理论。
In this note some new Frobenius type divisibility results are obtained for premodular categories. In particular, we extend Corollary 3.4 of [Yu20] from the settings of super-modular categories to arbitrary pseudo-unitary premodular categories.
We present an explicit formula computing morphism spaces in the stable category of a Frobenius algebra.
Affine oriented Frobenius Brauer categories
本文为任意 Frobenius 超代数 A 构造有向及仿射有向 Frobenius Brauer 范畴，并定义其在 gl_{m|n}(A) 超模范畴上的自然作用，统一推广一般线性与 queer 李超代数情形。
Cluster characters for 2-Calabi-Yau Frobenius extriangulated categories
本文定义带丛倾斜对象的 2-Calabi–Yau Frobenius 外三角范畴的丛特征标，统一了三角与正合两类范畴中的既有理论。
Algebraic Phase Theory II: The Frobenius Heisenberg Phase and Boundary Rigidity
本文提出代数相理论中弗罗贝尼乌斯海森堡相的刚性理论，通过代数方法证明其在有限弗罗贝尼乌斯环上的表示结构。
We introduce Nakayama functors for coalgebras and investigate their basic properties. These functors are expressed by certain (co)ends as in the finite case discussed by Fuchs, Schaumann, and Schweigert. This observation allows us to define Nakayama functors for Frobenius tensor categories in an intrinsic way. As applications, we establish the categorical Radford $S^4$-formula for Frobenius tensor categories and obtain some related results. These are generalizations of works of Etingof, Nikshych, and Ostrik on finite tensor categories and some known facts on co-Frobenius Hopf algebras.
We show that the bigroupoid of separable symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of finitely semi-simple Calabi-Yau categories are equivalent. To this end, we construct a trace on the category of finitely-generated representations of a symmetric, separable Frobenius algebra, given by the composite of the Frobenius form with the Hattori-Stallings trace. In order to put this result into perspective, we introduce the concept of the equivariantization of a 2-functor between bicategories endowed with a G-action. We prove that the equivalence between Froben…
Let $\Ascr,\Bscr$ be exact categories with $\Ascr$ karoubian and $M$ be an exact functor. Under suitable adjonction hypotheses for $M$, we are able to show that the direct factors of the objects of $\Ascr$ of the form $MY$ with $Y \in \Bscr$ make up a Frobenius category which allow us to define an $M$-stable category for $\Ascr$ only by quotienting. In addition, we propose a construction of an $M$-stable category for $\Ascr,\Bscr$ triangulated categories and $M$ a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck which links the two notions of $M$-stable cat…
We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gor…
Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\C$. Especially, for a $2$-Calabi-Yau extriangulated category $\C$ with a cluster structure, we describe the cluster substructure in the cotorsion pairs. For rooted cluster algebras arising from $\C$ with cluster tilting objects, we give a one-to-one correspondence between cotorsion pairs in $\C$ and certain pairs of their rooted cluster subalgebras which we call c…
Let ${\mathcal E}$ be a Frobenius category, ${\mathcal P}$ its subcategory of projective objects and $F:{\mathcal E} \to {\mathcal E}$ an exact automorphism. We prove that there is a fully faithful functor from the orbit category ${\mathcal E}/F$ into $\operatorname{gpr}({\mathcal P}/F)$, the category of finitely-generated Gorenstein-projective modules over ${\mathcal P}/F$. We give sufficient conditions to ensure that the essential image of ${\mathcal E}/F$ is an extension-closed subcategory of $\operatorname{gpr}({\mathcal P}/F)$. If ${\mathcal E}$ is in addition Krull-Schmidt, we give suffi…
We generalize the Frobenius-Schur theorem to $C^*$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^*$-algebras. Based on these definitions, we give another type of the Hopf-algebraic analogue of the Frobenius-Schur theorem, originally due to Linchenko and Montgomery. Also given are similar results for weak Hopf $C^*$-algebras, table algebras and compact quantum groups.
We explore the connection between the notion of Hopf category and the categorification of the infinite dimensional Heisenberg algebra via graphical calculus proposed by M.Khovanov. We show that the existence of a Hopf structure on a semisimple symmetric monoidal abelian category implies existence of a categorical action in the sense of Khovanov and thus leads to a strong categorification of this algebra.
We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.
Ideal $n$-cotorsion pairs in Frobenius extriangulated categories
研究Frobenius外三角范畴中理想n-余挠对与其稳定范畴中对应关系，证明两者一一对应，并建立完全余挠对与完全理想余挠对的对应，回答Fu等人的问题。
A geometric model for blocks of Frobenius kernels
本文基于 Steinberg 张量积公式的几何对应，定义并研究了 Langlands 对偶群的仿射 Grassmannian 上的反常层范畴，为代数群 Frobenius 核的表示块提供几何模型，并证明其具有射影、内射对象且为最高权范畴。
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariant…
We characterize Frobenius and separable monoidal algebra extensions $i: R\ra S$ in terms given by $R$ and $S$. For instance, under some conditions, we show that the extension is Frobenius, respectively separable, if and only if $S$ is a Frobenius, respectively separable, algebra in the category of bimodules over $R$. In the case when $R$ is separable we show that the extension is separable if and only if $S$ is a separable algebra. Similarly, in the case when $R$ is Frobenius and separable in a sovereign monoidal category we show that the extension is Frobenius if and only if $S$ is a Frobeniu…
Frobenius-Perron Dimensions of Conjugacy Classes and an Ito-Michler-Type Result in Modular Fusion Categories
本文研究有限群共轭类大小的算术条件在融合范畴中的类比，并为模融合范畴建立了Ito-Michler型结果。
A new characterization of silting subcategories in the stable category of a Frobenius extriangulated category
该研究给出Frobenius外三角范畴稳定范畴中silting子范畴的新刻画，建立其与原范畴共变有限子范畴的双射对应，推广了三角范畴的相关结果，并应用于同伦、导出等多类范畴。
Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras
本文构造了从子群H对应的Dijkgraaf-Witten范畴到G对应范畴的可分Frobenius幺半函子，并据此分类了刚性Frobenius代数，推广了Davydov-Simmons的étale代数分类至任意特征。
A refined multiplication formula in 2-Calabi-Yau Frobenius extriangulated categories
本文在Hom-有限2-Calabi-Yau Frobenius扩张三角范畴中，证明带簇倾斜对象的Wang-Wei-Zhang簇特征乘法公式，推广一维及稳定情形结果，并用最小余合冲中间项表达Auslander-Reiten网格冻结项，在Higgs范畴计算冻结重数获得主系数网格关系。
We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obt…
$\otimes$-Frobenius functors and exact module categories
本文研究了 ⊗-Frobenius 函子的性质及其在模块范畴中的行为，揭示了其与精确性、pivotality、unimodularity 和 sphericality 的关系，并给出了相关应用和结论。
Relative cluster categories and Higgs categories
本文将簇范畴构造推广至相对语境，证明了由左(n+1)-Calabi--Yau态射产生的Higgs范畴中存在稳定n-Calabi--Yau的n-cluster tilting对象，并应用于相对Ginzburg dg代数与高维Auslander代数。
Frobenius algebra objects in Temperley-Lieb categories at roots of unity
本文在根为单位的Temperley-Lieb范畴中定义了Frobenius结构，并展示了其Nakayama映射的阶为2，扩展了Frobenius代数的概念。
We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ i…
We associate a monoidal category $\mathcal{H}^λ$ to each dominant integral weight $λ$ of $\widehat{\mathfrak{sl}}_p$ or $\mathfrak{sl}_\infty$. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to $λ$. We show that, in the $\mathfrak{sl}_\infty$ case, the level $d$ Heisenberg algebra embeds into the Grothendieck ring of $\mathcal{H}^λ$, where $d$ is the level of $λ$. The categories $\mathcal{H}^λ$ can be viewed as a graphical calculus describing induction and restriction functors between categor…
We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius categories we consider are the categories of finitely generated Gorenstein projective modules over the singular Nakajima category associated to a Dynkin diagram and their standard Frobenius quotients. In particular, we are able to categorify all finite-type skew-symmetric cluster algebras with universal coefficients and finite-type Grassmannian cluster algebras. Along the way, we classify the standard Frobenius models of a certain famil…
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde…
The trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra W_{1+\infty}. This induces an action of W_{1+\infty} on symmetric functions.
A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group
本文将Stone-von Neumann定理推广至非酉且非不可约表示，基于du Cloux的思想建立了Heisenberg群与其中心表示间的光滑范畴等价，并将其应用于振子表示与退化Whittaker模型。
An update on Heisenberg and Kac-Moody categorification
本文针对A型表示论中从海森堡作用构造卡茨-穆迪2-范畴作用时第二个伴随难描述的问题，推导其显式公式，简化了现有理论。
Free monodromic Hecke categories and their categorical traces
本文提出了一种新的自由单值海克类别构造方法，并通过该方法证明了德利涅-卢斯特茨理论中的关键定理。
Deligne categories and representations of the finite general linear group, part 1: universal property
本研究刻画了参数为复数t的德利涅插值范畴Rep(GL_t(F_q))的态射空间，证明其生成对象为F_q-线性弗罗贝尼乌斯空间，且该范畴是此类维数t的对象生成的泛对称幺半范畴，还将在第二部分证明GL_∞(F_q)表示范畴的类似泛性质。
Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive self-adjoint Hopf algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the symmetric self-adjoint Hopf st…
Reduction of Frobenius extriangulated categories
本文提出 Frobenius 扩张三角范畴关于刚性子范畴的约化方法，推广 Iyama--Yoshino 约化，并用内部 Calabi--Yau 代数刻画关系，进而简洁证明饰带模式相关结果。
The spectral radius of matrix, also known as Frobenius-Perron dimension, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, find a way to calculate the Frobenius-Perron dimension of these algebras when they satisfy the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver …
We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (t…
We classify pointed fusion categories C(G, $ω$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their Frobenius-Schur indicators, their categorical Morita equivalence classes are distinguished by the set of the indicators and ribbon twists of their Drinfeld centers. In particular, the modular data are a complete invariant for the modular categories Z(C(G, $ω$)) for |G[< 32. We use the computer algebra package GAP and present codes for treating complex-valued gro…
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